A Constrained Optimal Control Approach to Smoothing Splines

This paper addresses smoothing spline estimation of complex functions subject to shape and/or dynamics constraints. Such estimation problems receive growing interest in engineering and statistics, particularly newly emerging areas such as systems biology. In this paper, we formulate the estimation problem as an optimal control problem subject to convex control constraints. By exploring techniques from convex and variational analysis, the existence and uniqueness of optimal solutions is established and explicit optimality conditions are obtained. It is shown that the optimality conditions are given in term of a two-point boundary value problem for a complementarity system. To compute an optimal solution, we formulate the optimality conditions as a B-differentiable equation. A nonsmooth Newton's method is exploited to solve this equation; global convergence of this method is established.